For any curve
(is nothing but the prolongation of a section of the trivial bundle
Then, the action for that curve is the integral of the composition
where
with fixed
(in the context of the variational bicomplex, the Lagrangian is a differential form
The trick to find
We can choose any vector field over
This curve gives us a
(By the way, we are deriving using the local chart connection).
If we consider the smooth function
its derivative respect to
Integrating the second term by parts:
And since the end points are fixed
And since this is valid for every
or more compactly
It is a straightforward calculation to check that Euler-Lagrange equations "behave in a good way" for change of variables.
This is generalized by the Euler operator: see the end of variational derivative#Some facts.
xournal_154
With the notation of the jet bundle
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Author of the notes: Antonio J. Pan-Collantes
INDEX: